To validate our approximation, we perform numerical simulations of this Hamiltonian Dysthe equation and compare them to computations
based on the full water wave system and another related Dysthe equation recently derived by [1]} in the same setting.
For a range of values of the vorticity, we examine the long-time dynamics of perturbed Stokes waves
and find very good agreement, thus providing a verification for both Dysthe models.
In particular, the performance of our Hamiltonian model is found to be quite satisfactory over the entire range considered.
We observe that the presence of vorticity clearly has an effect on the BF instability of Stokes waves on deep water.
Consistent with results from previous studies, a counter-propagating shear flow (negative vorticity) tends to enhance this instability
as it amplifies the growth rate and enlarges the instability region to higher sideband wavenumbers,
while a co-propagating current (positive vorticity) tends to stabilize it.
We hope this Hamiltonian Dysthe equation may serve as an efficient tool to study wave-current interactions in future applications.
As subsequent work, it would be of interest to extend the present method to the situation of constant finite depth with possibly surface tension.
For this problem, the reduction to normal form is expected to be significantly more complicated.
